The fee tends not to be directly related to the price; rather, the fee tends to be related to the rate of change of the price. Over a long enough time frame, then the price of bitcoin will tend towards stability. When it does, then the fees will tend towards stability as well.

When the price makes big swings up or down, then fees rise as lots of people scramble to either deposit coins to exchanges to sell, withdraw coins they've just bought, or move coins between exchanges so they can use margins, leverage, trade alts, whatever. When the price is stable, then the fees tend to level out as this surge of activity diminishes. Right now, for example, we are sitting at an all time high, and yet I just had a transaction confirm in the next block with a fee of 3 sats/vbyte. Given this, now would be a great time to drop say 0.02 BTC in to a new Lightning Channel for 3 sats/vbyte, and I can pay for 3 months worth of coffee for the fee of a couple of cents.